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On black hole microstates
•Introduction
•BH entropy
•Entanglement entropy
•BH microstates
Amos Yarom.
Ram Brustein.Martin Einhorn.
General relativity
22
2
sin000
000
00)/21(
10
000/21
r
rrM
rM
g
G=T =0
r=2M
r=0
Coordinate singularity
Spacetime singularity
1000
0100
0010
0001
g
Kruskal extension
MSinheMG
Mrt
MCosheMG
Mrx
Mr
Mr
4/2
)2(
4/2
)2(
2/
2/
22
2
sin000
000
000
000
r
r
h
h
g
Mrer
GMh 2/
332
MreMG
Mrxt 2/22
2
)2(
Previous coordinates:
rM2
t
x
r=2M
r=0
t=0
t=1/2
t=1
t=3/2
x
Entanglement entropy
21212
10,0
0000
02/12/10
02/12/10
0000
0,00,0
21 Trace
2/10
02/1
S=0
S=Trace (ln1)=ln2S=Trace (ln2)=ln2
All |↓22↓| elements
1 2
2
Finding 1
''00')'','(
DLdtExp ][00
(x,0)=(x)
00
x
t
’(x)’’(x)
Tr2 (’’’1(’1,’’1) =
1’1’’1 Exp[-SE] D
(x,0+) = ’1(x)(x,0-) = ’’1(x)
(x,0+) = ’1(x)2(x)(x,0-) = ’’1(x)2(x)
Exp[-SE] DD2
DLdtExp ][)'','(
(x,0+)=’(x)
(x,0-)=’’(x)
DLdtExp ][)'','(
(x,0+)=’(x)
(x,0-)=’’(x)
What does BH entropy mean?• BH Microstates
• Horizon states
• Entanglement entropy
√x
t
’1(x)
’’1(x)
’| e-H|’’
Kabbat & Strassler (1994), R. Brustein, M. Einhorn and A.Y. (to appear)
Finding 1
1’1’’1 Exp[-SE] D
(x,0+) = ’1(x)(x,0-) = ’’1(x)
MSinheMG
Mrt
MCosheMG
Mrx
Mr
Mr
4/2
)2(
4/2
)2(
2/
2/
Counting of microstates
(Conformal) field theoryCurved spacetime
Quantized gravity
4 L
String theory
AdS/CFT
SCFTNL 4
Ng YMs /4
AdS space CFT
Minkowski space
deSitterAnti deSitter
O
Z(b=0) Exp(OdV)=
YMR 4
Maldacena (1997)
YMR 4
SBH=A/4
SCFTNL 4
S=A/3
Semiclassical gravity:R>>’
Free theory: 0
S/A
1/R
AdS BH EntropyS. S. Gubser, I. R. Klebanov, and A. W. Peet (1996)
Anti deSitter +BH
AdS/CFT
CFT, T>0
What does BH entropy mean?• BH Microstates
• Horizon states
• Entanglement entropy
√
√
AdS BH
212
iii
EEEe
i
SCFTNL 4
AdS BH
AdS/CFT
CFTCFT, T=0CFT, T>0
?
|0
iii
E EEe i
11
0021 Trace
Maldacena (2003)
Generalization
)(00
0)(/10
00)(
rq
rf
rf
g
aSinhrgt
aCoshrgx
/)(
/)(
)('
2
)(
0
12
2
rfa
eCarg
r
drfa
Field theory
L
BH spacetime
f(r0)=0
)(00
0)(0
00)(
rq
rh
rh
g 22
12
1
)(
)(
txrg
feCrh
r
drfa
1’1’’1 Exp[-SE] D
(x,0+) = ’1(x)(x,0-) = ’’1(x)
’| e-H|’’
GeneralizationBH spacetime
HeTr 100
BH spacetime Field theory
L?
dHdd eTr 100
LΗ d
/2
2120 ii
i
E
d EEei
GeneralizationBH spacetime
HeTr 100
BH spacetime
Field theory
dHdd eTr 100
Field theory Field theory
LLH d
/2
2120 ii
i
E
d EEei
Summary
• BH entropy is a result of:– Entanglement– Microstates
• Counting of states using dual FT’s is consistent with entanglement entropy.
Entanglement entropy
121
0 aA a
2
)()( 21kk TrTr
S1=S2
Srednicki (1993)
00
,,,, ba
ba AbaA
ba
ba AbaA,,
*TAA
c
cc 00
,,,, ba
ba cAbaAc
,,b
bb AA
†AA
002Tr 001Tr
AdS/CFT (example)
dVOExpZ b 00 )( )(
0 )( Ib eZ
xdDDgI d 1
2
1)(
Witten (1998)
Massless scalar field in AdS An operator O in a CFT
0
DD
')'('
),( 0220
00 xdx
xxx
xcxx d
d
d
xx
xxcdI 2
00
'
)'()(
2)(
')'()',()(
2
1 00
0
xxddxxxGxExp
dVOExp
dd
)'()()',( xOxOxxG
dxx
cdxOxO 2
'
1
2)'()(
dVOExp 0
d
dd
xx
xxcdxxddxxxGx 2
0000
'
)'()(
2')'()',()(
2
1
d
xx
xxcd2
00
'
)'()(
2
Exp( )